CBO estimates that Boehner's amended bill would reduce deficits by $917 billion over 10 years. That's $17 billion more than the immediate debt ceiling increase of $900 billion that the bill would authorize.

Boehner's plan -- and a competing Democratic bill in the Senate -- are the only live bills this week that would increase the debt ceiling by Aug. 2. The ceiling must be raised by then, when the Treasury Department estimates it will no longer be able to guarantee it can pay all its bills without borrowing.

The House may vote on the amended bill on Thursday.

But even if the Boehner's bill passes the House, it has no chance in the Senate.

As the news broke that the amended bill got a better score from CBO, a letter signed by 53 senators (51 Democrats and 2 independents) indicated that they would not support it.

The CBO said the bulk of deficit savings under Boehner's amended bill -- $756 billion -- would result from caps on discretionary spending.

The other big chunk of savings -- $156 billion -- would come from reduced interest costs on the debt.

Almost as soon as Boehner proposed his bill on Monday, it came under fire from the most conservative members of his caucus and some conservative groups for not going far enough to reflect the principles of the Cut, Cap and Balance Act, which the House passed last week.

Cut, Cap and Balance would, among other things, cut total spending by $111 billion for fiscal year 2012. It would also require a balanced budget amendment to the Constitution that would cap total annual spending at 18%.

The spending caps in Boehner's bill would result in $25 billion in savings in the first year,and from there the annual savings would grow over time.

In addition, the Boehner bill would require that both chambers of Congress vote on a Balanced Budget Amendment but doesn't require that one be enacted.

The conservative Club for Growth also expressed its displeasure. "It cuts almost nothing immediately, it caps only discretionary spending, and it does not require passage of a balanced budget amendment." (Read: Betting on a U.S. default)

But the blowback rankled leadership and on Wednesday at a heated House GOP meeting Boehner reportedly warned members to "get your ass in line."

Boehner is in a tough spot. He recognizes that the debt limit must be raised to prevent the country from defaulting on its obligations. But he is also representing the will of his most conservative members, who have not yielded in their demands for large, immediate spending cuts as a condition for raising the debt ceiling.

That has put him at odds with President Obama, who has pushed for a debt reduction package that also includes a revenue component.